In filling machines of said type, use has previously been made of a method of approach which entails that, during the gas sterilisation stage, a sterilisation gas is supplied centrally in conjunction with the open end of each package. For reasons of process engineering, the sterilisation gas flow has been allowed to remain constant, regardless of the necessary movements in the filling machine. This is linked to the situation that the relevant technology is applicable in connection with filling machines of different types, such as also those of the tube filling- or carousel filling type. It has proved that, in certain cases, there may be a minor risk that the sterilisation gas which, in a previously known manner, is supplied to packages of this type in some cases does not reach the small areas and pockets of the package which are located least readily accessible in relation to the sterilisation agent supplying device. This applies particularly to such packages which display a relatively large ratio between their length and their main axis cross section. It has long been assumed that it is possible to reach these small areas and pockets of the packages located least readily accessible by maintaining a high constant gas mass flow in the supplied sterilisation gas. However, it has now been established that, above all in certain cases, there is a risk that the preselected level of sterilisation will not be achieved simply by this measure.